A Spectral Algorithm for 3-valued Function Equivalence Classification

Spectral techniques for Boolean and multiple-valued functions have been well studied and found to be useful in logic design and testing for conventional circuits. Spectral techniques also have potential application for reversible and quantum circuits. This paper addresses the classification of 3-valued functions into spectral translation equivalence classes. A transform algorithm is presented that determines the spectral translations to map a given function to the representative function for the equivalence class containing the given function. Using this algorithm we show, by exhaustive enumeration, that the 2-variable 3-valued functions partition into 11 equivalence classes. The number of 3-valued functions with 3 or more variables is very large, prohibiting exhaustive enumeration. We show that a search of 3-valued function 1-neighbourhoods yields 167,266 equivalence classes for 3 variables. The transform algorithm can be used for a higher number of variables to determine if two functions fall within the same equivalence class and, if they do, to find a sequence of spectral translations to map one to the other.